[seqfan] Concerning quasi-Carmichael numbers

Elijah Beregovsky elijah.beregovsky at gmail.com
Wed Feb 5 11:53:29 CET 2020

Hello, Seqfans!
In OEIS there’re sequences of quasi-Carmichael numbers (https://oeis.org/A257750). But they do not agree with each other on definition, it seems. The definition in A257750 is the following:

Quasi-Carmichael numbers are squarefree composites n with the property that for every prime factor p of n, p+b divides n+b positively with b being any integer besides 0.

But in most other sequences the terms correspond to “...p-b divides n-b...” Look, for example, at A029560. Those are “ Quasi-Carmichael numbers to base 3“ but at the same time “squarefree composites n such that prime p|n ==> p-3|n-3.” That disagreement is all over those sequences and it’s so confusing, that I at first even thought they talk about completely different sets called “quasi-Carmichael numbers”. 

Another thing is that some sequences call b “the base” and other “the order” of a quasi-Carmichael number. (compare A029560 and A029591).

I think, all this confusion grows from the paper of K. Bouallegue, O. Echi, R. G. E. Pinch (look in A029560) on “Korselt numbers”, that used a different terminology than OEIS. But their works is pretty much the only thing there is in the web concerning quasi-Carmichael numbers and all others cite this paper.
So I suggest this (read as: beg someone to do it): change the definition of quasi-Carmichael numbers to match that of Bouallegue et al., clean up their comments, change all words “order” to “base”, add some sequences concerning Korselt numbers (their definition permits b larger than n, so the quasi-Carmichael numbers are a subset of Korselt numbers) and link them to quasi-Carmichael pages. (I would have done this myself, but I’m limited to 3 submissions.)

Thanks in advance,

PS: All this I explained in the draft for A029591, so look there before you start making changes. And you could just reference this draft as a basis for changes. 

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